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Blobbed topological recursion: properties and applications

Published online by Cambridge University Press:  27 May 2016

GAËTAN BOROT  and
SERGEY SHADRIN
GAËTAN BOROT
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: gborot@mpim-bonn.mpg.de
SERGEY SHADRIN
Affiliation:
Korteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands. e-mail: s.shadrin@uva.nl
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Abstract

We study the set of solutions (ωg,n)g⩾0,n⩾1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 39 - 87
Copyright
Copyright © Cambridge Philosophical Society 2016 

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